Do the Math

Me, in class, explaining all this last night.

A friend of mine has often give me this good advice: Don't guess at anything, if you can help it.

I get the advice often, because I tend to forget it: eyeballing and seat-of-the-pants are much more my natural style. So when I had a request for 20 ounce mugs - and they had to be pretty-damn-close to 20 ounces, not "20-ounces, give-or-take," my first thought was to make a few mugs that looked like they would shrink to about the right size, write on the bottoms what the wet dimensions were, and see which one worked out best, and proceed from there. That approach will get the job done,  but maybe there is a better way?

Enter arithmetic! Or maybe this is algebra? Actually it's geometry, but it is algebraic geometry...? Or arithmetic geometry...?(OOPS SORRY, DERAIL. Back on track now.)  All of the necessary information to tell me how large this mug has to be is available.

Q: Okay, Google: How many cubic inches is 20 ounces of water?

A: 36.1 cubic inches

So far, so good. Now, so I need to make a mug with an interior space that will shrink to 36.1 cubic inches, plus a little so the liquid is not right up to the brim. How might one do that? Well, starting simply, the equation to determine the volume of a cylinder is

π r²(H) = V

where "H" is the height, and V the volume. Now let's put in the numbers that we know. Well, π is 3.141etc, etc. V is 36, plus a little, right? Plus how much? If I want a half-inch of space between the rim and the liquid, that number is 3.5 (which I arrived at by doing the above equation, but skipped that part so I wouldn't be repeating myself.) So, let's put in 39.6 for "V". To get the radius, we need to decide on a diameter, and after measuring a couple of mugs of different sizes, I decided on 3 inches - doesn't sound like much but is actually a pretty wide mug. That makes the radius 1.5. One-point-five squared is 2.25, so now we have all the numbers except one, H; this is the number our equation will give us. Like this:

3.141 x 2.25 x H = 39.6

Now we need to cancel out some of those numbers. We could say the same thing like this:

3.141 x H = 39.6 ÷ 2.25

...which equals 17.6. So now we've got

3.141 x H = 17.6

Cancelling out the π, now it looks like this:

H = 17.6  ÷ 3.141, which equals 5.6 and some change.

So our mug (assuming it was a perfect cylinder, which it's not, but we'll get to that) should, after firing, measure 5.6" tall by 3" in diameter; that's the interior space, 'cause that's where your coffee goes. On the interior. Usually. On a good day. So while I can easily measure the interior diameter with calipers, I'll need to tack on 1/8 inch in height for the bottom, and the clay that stays on the wheel.

My claybody shrinks something like 13%, but that is plus or minus 2%; and these mugs really shouldn't be too large, but they absolutely cannot be too small, so I'm gonna assume 15% shrinkage. (Also makes the math easier to do in my head, when I need to.) So now we have wet dimensions of 6.5 (adding the 1/8 inch and rounding down a hair) by 3.5 - again with a slight round up of .05 inches. 

Well, now we're cooking! If I start with this cylinder, and then add some curves, if I am careful to curve in at one spot about the same amount as I curve out at another, I should still arrive at my 20 ounce capacity. I will be guessing - let's say estimating, sounds nicer - how much clay to use, and then if it doesn't feel right - too clunky, too flimsy - adjusting up or down. I don't mind that, because that only takes a few minutes to work out, as opposed to having to wait a whole firing cycle. 
I am going with 1.75 pounds of clay. These mugs will be in a restaurant setting, so will be frequently washed, and will mostly serve beer, so users may be less than dainty with them. I am thinking a little thicker than usual might do well. On the other hand, I don't want a behemoth, since 20 ounces of beer is a little weighty all by itself (weighs exactly 20 ounces, in fact!) so I am only going a tiny bit thicker than usual. 

I wish I could make these mugs today, but alas, I have to go pile brick on  top of one another: I've promised myself the stack will be finished today so that weight - all 1000+ pounds of it - will be off my mind. And when it's done I will enjoy another kind of pie, as a reward. 

Yay math.